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To evaluate the integral using the indicated trigonometric substitution, we'll proceed as follows:

### Step 1: Identify the Substitution
Given \( x = 3 \tan(\theta) \), we need to calculate \( dx \) and substitute into the integral. 

- **Substitute \( x = 3 \tan(\theta) \):**
  \[
  dx = 3 \sec^2(\theta) \, d\theta
  \]

- **Simplify the expression under the square root:**
  \[
  x^2 + 9 = (3 \tan(\theta))^2 + 9 = 9 \tan^2(\theta) + 9 = 9 (\tan^2(\theta) + 1) = 9 \sec^2(\theta)
  \]
  So, \( \sqrt{x^2 + 9} = 3 \sec(\theta) \).

### Step 2: Substitute into the Integral
The original integral is:
\[
\int \frac{8x^3}{\sqrt{x^2 + 9}} \, dx
\]
Substitute \( x = 3 \tan(\theta) \) and \( dx = 3 \sec^2(\theta) \, d\theta \):
\[
\int \frac{8(3 \tan(\theta))^3}{3 \sec(\theta)} \cdot 3 \sec^2(\theta) \, d\theta
\]
Simplifying:
\[
\int \frac{8 \cdot 27 \tan^3(\theta)}{3} \cdot \sec(\theta) \, d\theta = 72 \int \tan^3(\theta) \sec(\theta) \, d\theta
\]

### Step 3: Simplify the Integral
Recall that \( \tan^3(\theta) = \tan(\theta) \cdot \tan^2(\theta) \) and \( \tan^2(\theta) = \sec^2(\theta) - 1 \):
\[
72 \int \tan(\theta)(\sec^2(\theta) - 1) \sec(\theta) \, d\theta
\]
Expanding:
\[
72 \int (\tan(\theta) \sec^3(\theta) - \tan(\theta) \sec(\theta)) \, d\theta
\]
This can be split into two integrals:
\[
72 \left( \int \tan(\theta) \sec^3(\theta) \, d\theta - \int \tan(\theta) \sec(\theta) \, d\theta \right)
\]

### Step 4: Evaluate the Integrals
1. **Integral of \( \tan(\theta) \sec(\theta) \):**
\[
\int \tan(\theta) \sec(\theta) \, d\theta = \frac{\sec^2(\theta)}{2} + C
\]

2. **Integral of \( \tan(\theta) \sec^3(\theta) \):** This requires a more advanced method of integration, typically using reduction formulas or rewriting in terms of sine and cosine. After evaluating, we find:

\[
\int \tan(\theta) \sec^3(\theta) \, d\theta = ...
\]
Continue the solution

Evaluate the integral using the indicated trigonometric substitution: ∫(8x^3/√(x^2+9)) dx, where x = 3tan(θ).

Learn how to solve the integral ∫(8x^3/√(x^2+9)) dx using trigonometric substitution with x = 3tan(θ). This detailed step-by-step solution covers key concepts like trigonometric identities and algebraic simplification. Suitable for advanced calculus students in Grades 11-12.

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